Solve y=c_{1}e^2x+c_{2}e^3x

$$y = c _ { 1 } e ^ { 2 x } + c _ { 2 } e ^ { 3 x }$$

Solve for c_1

$$c_{1}=-c_{2}e^{x}+\frac{y}{e^{2x}}$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$c_{1}e^{2x}+c_{2}e^{3x}=y$$

Subtract c_{2}e^{3x} from both sides.

$$c_{1}e^{2x}=y-c_{2}e^{3x}$$

The equation is in standard form.

$$e^{2x}c_{1}=y-c_{2}e^{3x}$$

Divide both sides by e^{2x}.

$$\frac{e^{2x}c_{1}}{e^{2x}}=\frac{y-c_{2}e^{3x}}{e^{2x}}$$

Dividing by e^{2x} undoes the multiplication by e^{2x}.

$$c_{1}=\frac{y-c_{2}e^{3x}}{e^{2x}}$$

Divide y-c_{2}e^{3x} by e^{2x}.

$$c_{1}=-c_{2}e^{x}+\frac{y}{e^{2x}}$$

Solve for c_2

$$c_{2}=\frac{y}{e^{3x}}-\frac{c_{1}}{e^{x}}$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$c_{1}e^{2x}+c_{2}e^{3x}=y$$

Subtract c_{1}e^{2x} from both sides.

$$c_{2}e^{3x}=y-c_{1}e^{2x}$$

The equation is in standard form.

$$e^{3x}c_{2}=y-c_{1}e^{2x}$$

Divide both sides by e^{3x}.

$$\frac{e^{3x}c_{2}}{e^{3x}}=\frac{y-c_{1}e^{2x}}{e^{3x}}$$

Dividing by e^{3x} undoes the multiplication by e^{3x}.

$$c_{2}=\frac{y-c_{1}e^{2x}}{e^{3x}}$$

Divide y-c_{1}e^{2x} by e^{3x}.

$$c_{2}=\frac{y}{e^{3x}}-\frac{c_{1}}{e^{x}}$$